Fuzzy System

1. Fuzzy System

2. Why Fuzzy • Based on intuition and judgment • No need for a mathematical model • Provides a smooth transition between members and nonmembers • Relatively simple, fast and adaptive • Less sensitive to system fluctuations • Can implement design objectives,(difficult to express mathematically), in linguistic or descriptive rules.

3. Applications Domain • Fuzzy Logic • Fuzzy Control • Neuro-Fuzzy System • Intelligent Control • Hybrid Control • Fuzzy Pattern Recognition • Fuzzy Modeling

4. Some Interesting Applications • Ride smoothness control • Camcorder auto-focus and jiggle control • Braking systems • Copier quality control • Rice cooker temperature control • High performance drives • Air-conditioning systems

5. Crisp Set Conventional or crisp sets are binary. An element either belongs to the set or doesn't. {True, false} {1, 0}

6. Universe (X) Crisp Set/Subset Subset A Subset B Subset C

7. e.g. On a scale of one to 10, how good was the dive? 10 9 9 9.5 Fuzzy Linguistic Variables • Examples of fuzzy measures include close, heavy, light, big, small, smart, fast, slow, hot, cold, tall and short.

8. Fuzzy Indicators • Can you distinguish between American and French person? • Some Rules: • If speaks English then American • If speaks French then French • If loves perfume then French • If loves outdoors then American • If good cook then French • If plays baseball then American

9. Fuzzy Indicators • Rules may give contradictory indicators {good cook, loves outdoors, speaks French} • The right answer is a question of a degree of association • Fuzzy logic resolves these conflicting indicators • Membership of the person in the French set is 0.9 • Membership of the person in the American set is 0.1

10. Fuzzy Versus Probability • Fuzzy  Probability • Probability deals with uncertainty and likelihood • Fuzzy logic deals with ambiguity and vagueness

11. Fuzzy Versus Probability • Fuzzy  Probability • Example #1 • Billy has ten toes. The probability Billy has nine toes is zero. The fuzzy membership of Billy in the set of people with nine toes, however, is nonzero.

12. #1 #2 Fuzzy Versus Probability Example #2 • A bottle of liquid has a probability of ½ of being rat poison and ½ of being pure water. • A second bottle’s contents, in the fuzzy set of liquids containing lots of rat poison, is ½. • The meaning of ½ for the two bottles clearly differs significantly and would impact your choice should you be dying of thirst. • 50% probability means 50% chance that the water is clean. • 50% fuzzy membership means that the water has poison. (cite: Bezdek)

13. A(x) 1 x 10 Crisp Membership Functions • Crisp membership functions are either one or zero. • e.g. Numbers greater than 10. A ={x | x>10}

14. B(x) 0 1 2 3 x Fuzzy Membership Functions • The set, B, of numbers near to 2 can be represented by a membership function

15. Fuzzy Subsets • A fuzzy set, A, is said to be a subset of B if • e.g.B = far and A=very far. • For example...

16. Fuzzy Sets Short Tall

17. Fuzzy Sets Short Tall Tall or Short?

18. Very Tall 7’ Tall 6’ Medium 5’ Short 4’ Very Short Fuzzy Measures

19. Medium Tall Very Tall Very Short Short 6’ 7’ 4’ 5’ Membership Function 1.0

20. 1.0 Medium Tall Very Tall Very Short Short 6’ 7’ 4’ 5’ Short Medium Tall Membership Function

21. Fuzzy Logic Operations Fuzzy union operation or fuzzy OR

22. Fuzzy Logic Operations Fuzzy intersection operation or fuzzy AND

23. Fuzzy Logic Operations Complement operation

24. Fuzzy Logic Operations Fuzzy union operation or fuzzy OR Fuzzy intersection operation or fuzzy AND Complement operation

25. B(x) A(x) 1 0 1 2 3 x 0 1 2 3 x AB(x) 0 1 2 3 x

26. B(x) A(x) 1 0 1 2 3 x 0 1 2 3 x A+B (x) 0 1 2 3 x

27. Fuzzification Rules Defuzzification Fuzzy System Basic Components Crisp input Output Result

28. Step1:Fuzzification • Fuzzifier converts a crisp input into a fuzzy variable. • Definition of the membership functions must • reflect the designer's knowledge • provide smooth transition between member and nonmembers of a fuzzy set • be simple to calculate • Typical shapes of the membership function are Gaussian, trapezoidal and triangular.

29. Example 1 • Assume we want to evaluate the health of a person based on his height and weight. • The input variables are the crisp numbers of the person’s height and weight. • Fuzzification is a process by which the numbers are changed into linguistic words

30. Medium Tall Very Tall Very Short Short 6’ 7’ 4’ 5’ Fuzzification of Height 1.0 VS = very short S = Short M = Medium etc.

31. Medium Heavy Very Heavy Very Slim Slim 250lb 300lb 150lb 200lb Fuzzification of Weight 1.0 VS = very slim S = Slim M = Medium etc.

32. Step 2 : Rules • Rules reflect expert’s decisions. • Rules are tabulated as fuzzy words • Rules can be grouped in subsets • Rules can be redundant • Rules can be adjusted to match desired results

33. Rules Function • Rules are tabulated as fuzzy words • Healthy (H) • Somewhat healthy (SH) • Less Healthy (LH) • Unhealthy (U) • Rule function f f= { U, LH, SH, H}

34. f U LH H SH 1 0.2 0.4 1 0.6 0.8 Decision Fuzzified Decision f= { U, LH, SH, H}

35. Fuzzy Rules Table

36. Step 3 : Calculate • For a given person, compute the membership of his/her weight and height • Example: • Assume that a person’s height is 6’ 1” • Assume that the person’s weight is 140 lbs

37. 1.0 Medium Tall Very Tall 0.7 Very Short Short 0.3 6’ 7’ 4’ 5’ height ={ VS, S, M, T , VT} height={ 0 0 0.7 0.3 0 } Membership of Height

38. 1.0 Medium Heavy Very Heavy 0.8 Very Slim Slim 0.2 250lb 300lb 150lb 200lb weight ={ VS, S, M, H , VH} Weight={ 0.8 0.2 0 0 0 } Membership of Weight

39. Step4:Activate Rules

40. Weight Height 0.8 0.2 Medium Heavy Very Heavy Very Short H SH LH U U Short SH H SH LH U 0.7 H LH U LH H U SH 0.3 H SH U Very Tall U LH H SH LH Substitute Membership Values

41. Perform min operation

42. Step5:Compute Decision Function f = {U,LH,SH,H}f = {0.3,0.7,0.2,0.2}

43. f LH 0.7 U H 0.3 SH 0.2 0.2 0.4 1 0.6 0.8 Decision Index Scaled Fuzzified Decision f = { U, LH, SH, H} f = { 0.3, 0.7, 0.2, 0.2}

44. Step6:Compute Final Decision • Use the fuzzified rules to compute the final decision. • Two methods are often used. - Maximum Method(not often used) - Centroid

45. Max Method • Fuzzy set with the largest membership value is selected. • Fuzzy decision: f={U, LH, SH, H} f={0.3, 0.7, 0.2, 0.2} • Final Decision(FD)=Less Healthy • If two decisions have same membership max, use the average of the two.

46. f LH 0.7 U H SH 0.3 0.2 0.2 0.4 1 0.6 0.8 Decision Index(D) Max Method

47. å m D m + m + D D ..... u u = = LH LH FD å m m + m + ..... u LH + + + 0.3 0.2 0 . 7 0.4 0 . 2 0.6 0 . 2 0.8 = = FD 0.4429 + + + 0 . 3 0 . 7 0 . 2 0 . 2 Centroid Method Crisp Decision Index (D) = 0.4429

48. f LH 0.7 U H SH 0.3 0.2 0.2 0.4 1 0.6 0.8 Decision Index(D) Fuzzified Decision Crisp Decision Index(D) is the centroid D=0.4429

49. f U LH H SH 1 0.75 0.25 0.2 0.4 1 0.6 0.8 Decision Index Fuzzy Decision Index Fuzzy Decision Index(D) 75% in Less Healthy group 25% in Somewhat Healthy group

50. Example 2 • Assume that we need to evaluate student applicants based on their GPA and GRE scores. • Let us assume that the decision should be Excellent (E), Very Good(VG), Good(G), Fair(F) or Poor(P) • An expert will associate the decision to the GPA and GRE score. They are then Tabulated.